Eigenvalues in terms of trace and determinant for matrices larger than 2 X 2

For a $2\times2$ matrix, $\operatorname{tr}$ and $\det$ are the matrix invariants that are the coefficients of the characteristic polynomial.
For a $3\times3$ matrix there are the same invariants and another one, given by $$ \frac{1}{2}\left[(\operatorname{tr}A)^{2}-\operatorname{tr}(A^{2})\right] $$ but expressing the eigenvalues in terms of invariant means to solve a cubic equation.

For higher dimensions there are other invariants, but solving a polynomial equation cannot be done by a general formula for $n\geq5$.

There is if you generalize in the correct manner. The characteristic equation $\lambda^n+\sum\limits_{i=0}^{n-1}c_i\lambda^i=0$ can be expressed with coefficients in terms of the trace and the determinant of the matrix, but as $n$ grows, this gets extremely laborious. Please see this Wikipedia article.

Of particular interest are $c_{n-1}=-\DeclareMathOperator{\tr}{tr} \tr(M)$ and $c_0=(-1)^n\det(M)$.